1. Field of Invention
The invention relates to a nonlinear conversion system and the method thereof. In particular, it relates to a nonlinear conversion system with precision mapping and the method thereof.
2. Related Art
Nonlinear conversions, such as the log convert, log add, square root, cosine function, and so on, have very wide applications. However, different systems may not be provided with the required nonlinear conversion operations. Therefore, some additional techniques along with the basic addition and multiplication operations are needed to achieve the goal.
The most common method to approximate the log convert is to restrict the search range in the domain [1˜2). For example, if one wants to computeY=log10X,then X is first converted:X=0.1x1x2 . . . xn*2−k=1.x1x2 . . . xn*2−(k+1) and its log in the base of 2 is evaluated:log2(X)=log2(1+x)−(k+1)where (1+x)ε[1˜2). Therefore, the logarithmic operation becomes a log in the domain of [1˜2) in the base of 2. Therefore, one first calculatesy=log2(1+x),and converts it back to a log in the base of 10Y=[y−(k+1)]·log102.This then gives the original value.
From the above derivation, one knows that the log function of any number X can be converted and mapped to a smaller domain. One first obtains the log function in the domain [1˜2) in the base of 2 (i.e. calculating y). The original log function is then computed by multiplying it with an appropriate coefficient. The search domain after such a conversion is restricted to [1˜2). Likewise, one only needs to search between log21˜log22 when doing approximations. The domain and complexity are thus greatly reduced. For example, in U.S. Pat. No. 5,951,629 proposed by Motorola, the domain [1˜2) is divided into several sections, as shown in FIG. 1. Different sections of the curve are approximated by polynomialsy=b0+b1(1+x)+b2(1+x)2.
To simplify the computational structure, the 2nd-order term is used as a compensation done by first estimating its approximate value for different inputs of x. Therefore, the actual formula should bey=b0+b1(1+x)+f2(x).The simplified calculation structure is shown in FIG. 1A.
Moreover, since multiplication involves more complexity, we sacrifice some precision to simplify the operation. Suppose each section of the curve can be approximated by a straight line:g=mix+bi 
If the approximate slope of some section is expressed in the base of 2 as mi=1.001011 . . . 2, then mi x is expressed as:
            m      i        ×    x    =      x    +          x      8        +          x      32        +          x      64      
If the approximate slope of another section is mj=0.1101001 . . . 2, then mj x is expressed as:
            m      j        ×    x    =            x      2        +          x      4        +          x      16        +          x      128      
Using the above-mentioned method, the approximate parameter mi of each section of the curve uses the first few effective 1 in the base of 2 (the first four 1 in this example) to simplify the multiplication operation of the first-order term. With the help of the pre-calculated bi, one is then able to obtain the approximate value. However, this method neglects the 2nd-order approximation to simplify the overall computation. The computation of the first-order term is achieved at the price of sacrificing some precision. Therefore, its ultimate precision of the approximation will be lower than using a complete multiplier.
To reduce the overall computation complexity, Motorola proposed another idea disclosed in U.S. Pat. No. 6,065,031. For the logrithmic operation in a limited input domain, the approximation for regions closer to straight lines is achieved by subtracting some offset, which we will not describe in further detail herein.
For the operation of square root, Ericsson employs the Newton-Raphson method in U.S. Pat. No. 6,389,443. However, this method has to compute continuously N times. Therefore, it is simplified to start from the point whose square root is 2N. Although the computing process is simplified, the precision is greatly sacrificed.
Therefore, for different nonlinear conversion systems whethere there is a universal mechanism that uses a simple structure and computational method to achieve the required precision is thus an important issue.